Effect of Hunting on Red Deer

Modelling Fecal Cortisol Metabolites

1, 2, 3, 4, 5

Dr. Nicolas Ferry - Bavarian National Forest (?) / Daniel Schlichting - StabLab

31 Jan 2025

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Goal

assess red deer stress response towards hunting activities

– on 41 individual collared red deer

– within Bavarian Forest National Park

– using FCMs

The Terminology

  • FCMs: Faecal Cortisol Metabolites - a non-invasive method to measure stress through faecal samples

  • collared deer: red deer wearing a GPS-collar, which provides hourly location information

  • Euclidean Distance: Also known as \(L^2\) Distance. Reduces to Pythagorean Theorem for two Dimensions: \[d_{x,y} = ((x_1 - y_1)^2 + (x_2 - y_2)^2)^\frac{1}{2} \\ x,y \in \mathbb{R}^2 \]

The FCMs

  • FCM values do not represent stress level when defecating
  • FCMs show the stress levels of previous hours1
  • Reason: gut retention time

The Approach

  • model FCM levels on spatial and temporal distance to hunting activities

  • Expectation: FCM levels higher when closer in time and space

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Datasets

  • Movement: contains the location and datetime of the 41 collared deer in the period Feb 2020 - Feb 2023. In total approx. 740 000 observations1

  • Hunting Events: contains location and date of hunting events in the National Park - in total 1270 events, 890 of them with full timestamp

  • FCM Stress: contains information of 809 faecal samples, including:
    – the location of the sample
    – the time of sampling
    – the DNA-matched collared deer
    – the time when the deer was at the location

  • Reproduction Success: observations of 16 collared deer on:
    – if they were pregnant in one year
    – if they were accompanied by a calf in one year

The Fused Data

We introduce 4 Parameters:

  • Gut Retention Time (GRT) low: The minimum amount of hours, a Stress Event can appear before Defecation Time

  • Gut Retention Time (GRT) high: The maximimum amount of hours, a Stress Event can appear before Defecation Time

  • Distance Threshold: The maximum spatial Distance of a Deer to a given Hunting Event to be considered

  • Proximity Criterion: We consider either the Hunting Event closest in Space or in Time to be the most relevant

The Fused Data

Compute Temporal Distances

The Fused Data

Interpolate Movements

The Fused Data

Compute Spacial Distances

The Fused Data

Add Pregnancy Data

The Fused Data

Identify Events

The Fused Data

Finish Datasets

We suggest eight different Datasets for Modelling

DataSet GRT low GRT high Distance Threshold Proximity Criterion
1 19 50 10 last
2 14 50 10 last
3 19 50 20 last
4 14 50 20 last
5 19 50 10 nearest
6 14 50 10 nearest
7 19 50 20 nearest
8 14 50 20 nearest

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

A Generalized Additive Model

  • \(FCM_i \sim \mathcal{N}(\mu_i, \sigma^2)\)

  • Identity Link: \(E(FCM_i) = \mu_i = \eta_i\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

  • \(FCM_i \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_i})\)

  • For better Interpretability we use the Log-Link: \(E(FCM_i) = \mu_i = exp(\eta_i)\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

A Generalized Additive Model

B Generalized Additive Mixed Model

  • \(FCM_{i\,j} \sim \mathcal{N}(\mu_{i\,j}, \sigma^2)\)

  • Identity Link: \(E(FCM_{i\,j}) = \mu_{i\,j} = \eta_{i\,j}\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_{i\,j} = \beta_0 + \beta_1\,Pregnant_{i\,j} +\\ \beta_2\,Number\,Other\,Hunts_{i\,j} + f_1(Time\,Diff_{i\,j}) + \\ f_2(Distance_{i\,j}) + f_3(Sample\,Delay_{i\,j}) + f_4(Day\,of\,Year_{i\,j}) \end{gathered} \end{equation} \] with: \(\gamma_j \overset{\mathrm{iid}}{\sim} \mathcal{N}(0, \sigma_{\gamma}^2)\)

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

  • \(FCM_{i\,j} \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_{i\,j}})\)

  • For better Interpretability we use the Log-Link: \(E(FCM_{i\,j}) = \mu_{i\,j} = exp(\eta_{i\,j})\)

  • Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_{i\,j} = \beta_0 + \beta_1\,Pregnant_{i\,j} +\\ \beta_2\,Number\,Other\,Hunts_{i\,j} + f_1(Time\,Diff_{i\,j}) + \\ f_2(Distance_{i\,j}) + f_3(Sample\,Delay_{i\,j}) + f_4(Day\,of\,Year_{i\,j}) + \\ \gamma_j \end{gathered} \end{equation} \]

with: \(\gamma_j \overset{\mathrm{iid}}{\sim} \mathcal{N}(0, \sigma_{\gamma}^2)\)

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

B Generalized Additive Mixed Model

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Conclusion

  • Not enough observations after the datafusion left for robust modelling

Discussion

  • How to define space-time-distance?